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As smart contract platforms autonomously manage billions of dollars of
capital, quantifying the portfolio risk that investors engender in these
systems is increasingly important. Recent work illustrates that Proof of Stake
(PoS) is vulnerable to financial attacks arising from on-chain lending and has
worse capital efficiency than Proof of Work (PoW) \cite{fanti_pos_econ}.
Numerous methods for improving capital efficiency have been proposed that allow
stakers to create fungible derivative claims on their staked assets. In this
paper, we construct a unifying model for studying the security risks of these
proposals. This model combines birth-death Pólya processes and risk models
adapted from the credit derivatives literature to assess token inequality and
return profiles. We find that there is a sharp transition between 'safe' and
'unsafe' derivative usage. Surprisingly, we find that contrary to
\cite{fanti2019compounding} there exist conditions where derivatives can
\emph{reduce} concentration of wealth in these networks. This model also
applies to Decentralized Finance (DeFi) protocols where staked assets are used
as insurance. Our theoretical results are validated using agent-based

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